Improved Iterative MIMO Signal Detection Accounting for ...of multicarrier systems based on orthogonal frequency-division multiplexing (OFDM). To deal with imperfect channel estimation,

3154 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009

Improved Iterative MIMO Signal DetectionAccounting for Channel-Estimation Errors

Seyed Mohammad-Sajad Sadough, Member, IEEE, Mohammad-Ali Khalighi, Senior Member, IEEE,and Pierre Duhamel, Fellow, IEEE

Abstract—For the case of imperfect channel estimation, wepropose an improved detector for multiple-input–multiple-output(MIMO) systems using an iterative receiver. We consider twoiterative detectors based on maximum a posteriori and softparallel-interference cancellation and propose, for each case,modifications to the MIMO detector by taking into account thechannel-estimation errors. Considering training-based channelestimation, we present some numerical results for the caseof Rayleigh block fading, which demonstrate an interestingperformance improvement, compared with the classically usedmismatched detectors. This improvement is obtained without aconsiderable increase in receiver complexity. It is more advan-tageous when few pilots are used for channel estimation. Thus,the proposed improved detectors are of particular interest underrelatively fast-fading conditions, where it is important to reducethe number of pilots to avoid a significant loss in data rate.

Index Terms—Channel-estimation errors, improved maximuma posteriori (MAP) detection, mismatched detection, multiple-input–multiple-output (MIMO), soft parallel-interference cancel-lation (soft-PIC) detection.

NOTATION

Upper case bold symbols matrices;lower case bold symbols vectors;IN N × N identity matrix;Ex[.] expectation with respect to the ran-

dom vector x;det{.} matrix determinant;tr{.} matrix trace;|.| absolute value;‖.‖ Frobenius norm;(.)T vector transpose;(.)† Hermitian transpose;(.)∗ complex conjugate.

Manuscript received January 27, 2008; revised October 28, 2008 andJanuary 17, 2009. First published February 27, 2009; current version publishedAugust 14, 2009. The review of this paper was coordinated by Prof. E. Bonek.

S. M.-S. Sadough is with the Department of Telecommunications, Facultyof Electrical and Computer Engineering, Shahid Beheshti University, Tehran1983963113, Iran (e-mail: [email protected]; [email protected]).

M.-A. Khalighi is with the École Centrale Marseille, 13451 Marseille Cedex20, France, and also with Institut Fresnel, 13397 Marseille Cedex 20, France.

P. Duhamel is with the National Scientific Research Center/Laboratory ofSignals and Systems–Supélec, 91192 Gif sur Yvette Cedex, France.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2009.2016103

I. INTRODUCTION

I T is well known that multiple-input–multiple-output(MIMO) systems can provide very high spectral efficiencies

in a rich scattering propagation medium [1]. They are, hence, apromising solution for high-speed spectrally efficient reliablewireless communication. One important issue regarding theimplementation of MIMO systems is the channel estimationwhen coherent signal detection is performed at the receiver [2].To obtain the channel-state information at the receiver (CSIR),one usually used approach consists of sending some knowntraining (also called pilot) symbols from the transmitter, basedon which the receiver estimates the channel before proceedingto the detection of data symbols. This method of obtainingthe CSIR is usually called pilot-symbol-assisted modulation(PSAM) [3]. Obviously, due to the finite number of pilotsymbols and noise, in practice, the receiver can only obtain animperfect (and possibly very bad) estimate of the channel.

A rich literature exists on the impact of imperfect channelestimation on the performance of communication systems thatemploy multiple antennas. For a MIMO system that uses PSAMfor channel estimation, Garg et al. showed in [4] that, tocompensate for the performance degradation due to imperfectchannel estimation, the number of receive antennas should beincreased. In [5], the authors investigated the effect of imperfectchannel estimation on space-time (ST) decoding and showedthat the classical maximum likelihood (ML) detector, whichwas derived for the case of perfect CSIR, becomes largelysuboptimal in the presence of channel-estimation errors. Onesimilar investigation was carried out in [6] and [7] in the caseof multicarrier systems based on orthogonal frequency-divisionmultiplexing (OFDM).

To deal with imperfect channel estimation, one suboptimalapproach, which is known as mismatched detection, consists ofusing the channel estimate in the detection part as if it was aperfect estimate. It is shown, for instance, in [8] that this schemegreatly degrades the detection performance in the presence ofchannel-estimation errors.

As an alternative to the aforementioned mismatched ap-proach, Tarokh et al. [8] and, recently, Taricco and Biglieri [5]proposed an improved ML detection metric under imperfectCSIR and used it in the standard Viterbi algorithm. Recently,in [9], it has been shown that, compared with the mismatchedML, the improved ML metric can increase the achievableoutage rates of MIMO-OFDM systems, in particular when onlya few training symbols are devoted for channel estimation.Similar observations are reported in [10]–[12] for the case ofsingle-antenna OFDM systems.

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SADOUGH et al.: IMPROVED ITERATIVE MIMO SIGNAL DETECTION FOR CHANNEL-ESTIMATION ERRORS 3155

Here, we consider iterative (turbo) detection at the receiver,which is an efficient technique when channel coding is used[13], [14]. This scheme is essentially composed of a MIMOdetector (also called a demapper) and a soft-input-soft-output(SISO) channel decoder that exchange soft information witheach other through several iterations.

One practical concern for the implementation of turbo de-tectors for MIMO systems is the receiver complexity, which ismainly dominated by that of the MIMO demapper. For instance,in the case of maximum a posteriori (MAP) detection [15],which is the optimal solution under perfect CSIR in the senseof bit error rate, this complexity exponentially grows with thenumber of transmit antennas and the signal constellation size.Thus, suboptimal detection techniques are often preferred toMAP detection. One interesting suboptimal detector is thatbased on soft parallel-interference cancellation (soft-PIC) andlinear minimum mean-square error (MMSE) filtering. Thisscheme was first proposed in [16] in the context of multiuserdetection and was later applied to MIMO systems in [17] and[18], for instance.

Our aim in this paper is to propose an improved iterative de-tector that takes into account the imperfect channel estimationobtained via training sequences, i.e., using the PSAM approach.To this end, we propose a Bayesian framework based on thea posteriori probability density function (pdf) of the perfectchannel, which is conditioned on its estimate. This generalframework enables us to formulate any detector by consideringthe average, over the channel uncertainty, of the detector’s costfunction that would be applied in the case of perfect channelknowledge. As we will see, the improved ML metric in [5]becomes a special case of our general framework.

First, by properly modifying the soft values at the output ofthe MAP detector, we reduce the impact of channel uncertaintyon the SISO decoder performance. Second, we propose animproved soft-PIC detector by taking into account the channel-estimation errors in the formulation of the instantaneous linearMMSE filter, as well as in the interference cancellation part.

The outline of this paper is given as follows. In Section II,we describe our MIMO channel model and our main assump-tions concerning data transmission and channel estimation.In Section III, we introduce a general Bayesian frameworkfor improved detection under imperfect channel estimation. InSection IV, we provide the formulation of MAP and soft-PICdetectors in the case of perfect CSIR. Using this material inSection V, we propose improved detectors under imperfectCSIR for the two cases of turbo-MAP and turbo-PIC detection.Section VI illustrates, via simulations, a comparative perfor-mance study of the proposed detectors with the classical mis-matched detector. The convergence analysis of turbo-PIC basedon mismatched and improved soft-PIC detectors is provided inSection VII. Finally, Section VIII concludes this paper.

II. SYSTEM MODEL AND CHANNEL ESTIMATION

A. MIMO Fading Channel

We consider a single-user MIMO system with MT transmitand MR receive antennas (MR ≥ MT ), which transmits overa frequency-nonselective channel, and we refer to it as an

MT × MR MIMO channel. At the transmitter, we employ thebit-interleaved coded modulation (BICM) scheme, which isknown to be a simple efficient method for exploiting channeltime selectivity [19]. The binary data sequence is encodedby a nonrecursive nonsystematic convolutional (NRNSC) codebefore being interleaved by a quasirandom interleaver. Theoutput bits are gathered in subsequences of B bits and aremapped to complex multilevel quadrature amplitude modu-lation (M = 2B) symbols s before being passed to the STencoder.

In what follows, unless otherwise mentioned, we consider theST coding in its simplest form, i.e., just spatial multiplexing ofdata symbols, which is also known as the Vertical Bell Labo-ratories Layered Space-Time (V-BLAST) scheme. We presentour model for this simple case, and we shall generalize it laterto the case of an arbitrary ST code.

Let us denote by xk the MT × 1 vector of transmitteddata symbols (after the ST encoder) at a sampling time k.Considering the simple spatial multiplexing, we use sk insteadof xk. Assuming a frame of symbols that correspond to Lchannel uses, which are transmitted over the channel matrix H,the received signal vector yk of dimension MR × 1 is given by

yk = Hsk + zk, k = 1, . . . , L (1)

where sk is the MT × 1 vector of transmitted symbol with

average power EsΔ= (1/MT )E[tr{sks†

k}]. We assume thatthe entries {H}i,j of the random matrix H are independentidentically distributed (i.i.d.) zero-mean circularly symmetriccomplex Gaussian (ZMCSCG) random variables. Thus, thechannel matrix H is distributed as H ∼ CN (0, IMT

⊗ ΣH),where

CN (0, IMT⊗ ΣH)

=1

πMRMT det{ΣH}MTexp

{−tr

{HΣ−1

H H†}} . (2)

Here, CN (.) denotes complex Gaussian distribution, ⊗ standsfor the Kronecker product, and ΣH is the MR × MR covari-ance matrix of the rows of H. With our assumptions of i.i.d.channel, ΣH is a diagonal matrix with equal diagonal entriesσ2

h. The noise vector zk is assumed to be a ZMCSCG random

vector with covariance matrix ΣzΔ= E(zkz†

k) = σ2zIMR

. BothH and zk are assumed to be ergodic and stationary randomprocesses.

We consider the block fading model for the channel timevariations. Thus, channel coefficients are assumed to be con-stant during a block of symbols and change to new independentvalues from one block to another. We assume that each frameof data symbols corresponds to Nc independent fading blocks.Notice that Nc = 1 returns to the quasistatic channel model.

B. Pilot-Based Channel Estimation

To estimate the MIMO transmission channel matrix H at thereceiver, which corresponds to each fading block, we send anumber of pilot symbols in addition to the ST-encoded datasymbols. We devote a number of NP channel uses to the



transmission of pilot vectors sP,i, (i = 1, . . . , NP ) for eachone of Nc fading blocks. Considering a given fading block,let us constitute the (MT × NP ) matrix SP by stacking inits columns the pilot vectors, i.e., SP = [sP,1|, . . . , |sP,NP

].According to (2), during a given channel training interval, wereceive

YP = HSP + ZP . (3)

The definition of YP and ZP is similar to that of SP . We denoteby EP the average power of the training symbols defined as

EPΔ=

1NP MT

tr{SP S†

P

}. (4)

The least squares estimate of H is obtained by minimizing‖YP − HSP ‖2 with respect to H, which coincides here withthe ML estimate [20], i.e.,

HML = YP S†P

(SP S†

P

)−1

. (5)

Let us denote by E the matrix of the estimation errors. We have

HML = H + E, with E = ZP S†P

(SP S†

P

)−1

. (6)

It is known that the best channel estimate is obtained with mutu-ally orthogonal training sequences, which result in uncorrelatedestimation errors. In other words, we should choose SP withorthogonal rows such that

SP S†P = NP EP IMT

. (7)

This way, the jth column Ej of E has the covariance matrixΣE , i.e.,

ΣE = E

[Ej E†

j

]= σ2

E IMR, where σ2

E =σ2

z

NP EP. (8)

Thus, based on (6), the conditional pdf of HML, given H, caneasily be expressed as

p(HML|H) = CN (H, IMT⊗ ΣE) (9)

where CN (Λ,Ξ) denotes a complex Gaussian distribution withmean Λ and the covariance matrix Ξ. Using the pdf of Hand (HML|H) from (2) and (9), we can derive the posteriordistribution of the perfect channel matrix, conditioned on itsML estimate, as follows (see Appendix I-A):

p(H|HML) = CN (ΣΔHML, IMT⊗ ΣΔΣE) (10)

where

ΣΔ = ΣH(ΣE + ΣH)−1. (11)

Under the assumptions of mutually orthogonal pilot sequencesand i.i.d. channel coefficients, we have ΣΔ = δ IMR

, and

p(H|HML) = CN(δHML, δσ2

E IMT⊗ IMR

)(12)

where

δ =σ2

h

(σ2h + σ2

E). (13)

The availability of the estimation error distribution is aninteresting feature of pilot-assisted channel estimation that weused to derive the posterior distribution (10). For simplicity, wedo not specify hereafter the subscript ML for H.

III. DETECTOR DESIGN IN THE PRESENCE OF

CHANNEL ESTIMATION ERRORS

Consider the model (1) and denote by f(yk, sk,H) the costfunction that would let us decide in favor of a particular sk atthe receiver if the channel was perfectly known. Under a pilot-based channel estimation characterized by the posterior pdf of(10), we propose a detector based on the minimization of a newcost function, which is defined as

f(yk, sk, H) =∫H

f(yk, sk,H)p(H|H)dH

= EH|H

[f(yk, sk,H)

∣∣H](14)

where we have averaged the cost function f over all realiza-tions of the unknown channel H conditioned on its availableestimate H by using the distribution (10). We note that thedetector that minimizes (14) is an alternative to the suboptimalmismatched detector that is based on the minimization ofthe cost function f(yk, sk, H). This latter value is obtainedby using the estimated channel H in the same metric thatwould be applied if the channel was perfectly known, i.e.,f(yk, sk,H). The proposed approach in (14) differs from themismatched detection on the conditional expectation E

H|H [.],which provides a robust design by averaging the cost func-tion f(yk, sk,H) over all realizations of channel-estimationerrors.

Consider the problem of detecting the symbol vector sk

from the observation model (1) in the ML sense, i.e., themaximization of the likelihood function, which we denote byW (yk|H, sk). It is well known that under i.i.d. Gaussian noise,the ML detection of sk leads to minimizing the Euclideandistance metric DML. We have

skML(H) = arg min

sk∈CMT ×1{DML(sk,yk,H)} (15)

with

DML(sk,yk,H) Δ= − log W (yk|sk,H) ∝ ‖yk − Hsk‖2

(16)

where ∝ means “is proportional to,” and C is the set of complexnumbers.

As an alternative to mismatched detection, we derive, inwhat follows, a modified likelihood criterion by averagingW (yk|H, sk) over all estimation errors. The modified like-lihood function that we denote by W (yk|H, sk) is obtained



Fig. 1. Structure of the iterative MIMO receiver.

by using the pdf (10) and the formulation provided in (14).We have

W (yk|H, sk) =∫

H∈CMR×MT

W (yk|H, sk)p(H|H) dH

= EH|H

[W (yk|Hk, sk)

∣∣H]. (17)

After some algebraic manipulations, which are provided inAppendix I-B, we obtain

W (yk|H, sk) = CN (mM,ΣM) (18)

where

mM = δHsk ΣM = Σz + δΣε‖sk‖2. (19)

Now, using (16), we can obtain the improved ML decisionmetric in the presence of imperfect channel estimation, whichwe denote by DM(sk,yk, H), i.e.,

DM(sk,yk, H) Δ= − log W (yk|H, sk)= MR log π

(σ2

z + δσ2E‖sk‖2

)+

‖yk − δHsk‖2

σ2z + δσ2

E‖sk‖2. (20)

Note that the metric independently proposed in [5] and [8] forthe case of ML detection can, in fact, be considered as a specialcase of the general framework (14), which leads to (20).

We note that, under near-perfect CSIR, which is obtainedwhen the number of pilot symbols tends to infinity, i.e.NP → ∞, we have

limNP →∞

DM(sk,yk, H)

DML(sk,yk, H)= 1. (21)

Consequently, the improved metric of (18) becomes equivalentto the mismatched metric for negligible estimation errors, i.e.,for σ2

E → 0.

IV. ITERATIVE MIMO DETECTION UNDER PERFECT CSIR

At the receiver, we perform iterative symbol detection andchannel decoding. As shown in Fig. 1, the receiver principallyconsists of a MIMO detector (also called demapper) and a SISOchannel decoder that exchange extrinsic soft information with

each other. Here, we consider this soft information in the formof log-likelihood ratio (LLR).

The SISO decoder is based on the Max-Log-MAP algorithm,as described in [21] and [22]. We briefly present, in the follow-ing sections, the principle of MIMO detection based on twoapproaches of MAP and soft-PIC, assuming that perfect CSIRis available at the receiver. This step constitutes the bases forthe next section, where we present improved detectors for thecase of imperfect CSIR.

A. MAP Detection

Let dmk be the mth (m = 1, 2, . . . , BMT ) bit that corre-

sponds to the symbol vector sk, transmitted at the kth timeslot. We denote by L(dm

k ) the LLR of the bit dmk at the output

of the MIMO detector. Conditioned to perfect CSIR, L(dmk ) is

given by

L (dmk ) = log

Pdem (dmk = 1|yk,H)

Pdem (dmk = 0|yk,H)

(22)

where Pdem(dmk |yk,H) denotes the probability of transmission

of dmk . Let S be the set of all possibly transmitted symbol

vectors sk. We partition S into Sm0 and Sm

1 , for which the mthbit of sk is equal to “0” or “1,” respectively. We have

L (dmk ) = log

∑sk∈Sm

1

e−DML(sk,yk,H)BMT∏n=1n �=m

P 1dec (dn

k )

∑sk∈Sm

0

e−DML(sk,yk,H)BMT∏n=1n �=m

P 0dec (dn

k )(23)

where P 1dec(d

nk ) and P 0

dec(dnk ) are a priori information that

comes from the SISO decoder. The summation in (23) istaken over the product of the likelihood W (yk|sk,H) =exp{−DML(sk,yk,H)}, given a symbol vector sk, and the apriori probability on this symbol (the term

∏Pdec), which is

fed back from the SISO decoder at the previous iteration. Inthis latter term, the a priori probability of the bit dm

k itself hasbeen excluded to respect the exchange of extrinsic informationbetween the channel decoder and the demapper. In addition,note that this term assumes independent coded bits dn

k , which istrue for random interleaving of large size. At the first iteration,where no a priori information is available on bits dn

k , theprobabilities P 0

dec(dnk ) and P 1

dec(dnk ) are set to 1/2.

B. Soft-PIC Detection

The computational complexity of the MAP detector becomesprohibitively large for large-sized signal constellations and/orfor a large number of transmit antennas, because each of thesets Sm

1 and Sm0 in (23) contains 2(BMT −1) vectors sk. For such

cases, the suboptimal soft-PIC detector would make a goodcompromise between complexity and performance [17], [23].In what follows, we recall the formulation of soft-PIC.

The general block diagram in Fig. 1 still applies to the turbo-PIC detector. Here, to detect a symbol transmitted from a givenantenna, we first make use of the soft information availablefrom the SISO channel decoder to reduce and, hopefully, cancel



the interfering signals that arise from other transmit antennas.At the first iteration, where this information is not available, weperform a classical MMSE filtering.

Let us consider the transmitted vector sk = [s1k, . . . , sMT

k ]T

at time k and assume that we are interested in the detection ofits ith symbol si

k. We start by evaluating the parameters sjk and

σ2sj

k

for the interfering symbols sjk, j �= i from the SISO decoder

as follows:

sjk = E

[sj

k

]=

2B∑j=1

sjkP

[sj

k

](24)

σ2sj

k

= E

[∣∣∣sjk

∣∣∣2] =2B∑j=1

∣∣∣sjk

∣∣∣2 P[sj

k

](25)

where P [sjk] is the probability of the transmission of sj

k andis evaluated using the probabilities Pdec(d

j,nk ) at the decoder

output. We have

P[sj

k

]= K

B∏n=1

Pdec

(dj,n

k

)where K is a normalization factor. We further introduce thefollowing definitions. Hi is the (MR × (MT − 1) matrix con-structed from H by discarding its ith column, i.e., hi. We alsodefine the (MT − 1) × 1 vectors as

sik

Δ=[s1

k, s2k, . . . , si−1

k , si+1k , . . . , sMT

k

]T

sik

Δ=[s1

k, s2k, . . . , si−1

k , si+1k , . . . , sMT

k

]T

where sjk are estimated in (24).

Now, given the received signal vector yk, a soft interferencecancellation is performed on yk to detect the symbol si

k bysubtracting to yk the estimated signals of the other transmitantennas as

yik

= yk − Hisik = his

ik + His

ik − His

ik + zk

for i = 1, . . . ,MT . (26)

Except under perfect prior information on the symbols, whichleads to sj

k = sjk, there remains a residual interference in yi

k. To

further reduce this interference, an instantaneous linear MMSEfilter wi

k is applied to yik

to minimize the mean-square value ofthe error ei

k defined as

eik = si

k − rik (27)

where the filter output rik is equal to

rik = wi

kyik. (28)

Here, wik is obtained as

wik = arg min

wik∈CMR

Esk,zk

[∣∣∣sik − wi

kyik

∣∣∣2] . (29)

By invoking the orthogonality principle [24], the coefficients ofthe MMSE filter wi

k are given by

wik = h†

i

[hih

†i +

Hi(Λk,i − Λk,i)H†i

σ2si

k

+σ2

z

σ2si

k

IMR

]−1

(30)

where

Λk,i = E

[si

ksik†]

≈ diag(

E

[∣∣s1k

∣∣2] , . . . , E[∣∣si−1

k

∣∣2]E

[∣∣si+1k

∣∣2] , . . . , E

[∣∣∣sMT

k

∣∣∣2])Λk,i = si

ksik

†

≈ diag(∣∣s1

k

∣∣2 , . . . ,∣∣si−1

k

∣∣2 ,∣∣si+1

k

∣∣2 , . . . ,∣∣∣sMT

k

∣∣∣2) .

Note that the off-diagonal entries in Λk,i and Λk,i have beenneglected to reduce the complexity without causing significantperformance loss [18].

At the first decoding iteration, we have no prior informationavailable on the transmitted data, i.e., Λk,i = σ2

sik

IMT −1, and

Λk,i = 0MT −1. Consequently, (30) reduces to

wik = h†

i

[HH† +

σ2z

σ2si

k

IMR

]−1

(31)

which is no more than the linear MMSE detector for sik.

Before being passed to the SISO decoder, the detectedsymbols rk at the output of the MMSE filter in (28) shouldbe converted to LLR. This method is done by assuming aGaussian distribution for the residual interference after soft-PICdetection. Details on the LLR conversion can be found in [16].For brevity, the discussion will be presented only for the caseof the improved detector in Section V-B.

1) Simplified Soft-PIC Detector: It is clear from (30) and(38) that the MMSE filter coefficients change as a functionof time and must be recomputed for each time symbol. Thiscondition motivates us to propose here a simplified version ofthe soft-PIC detector to further reduce the receiver complexity.

Consider the filter expression (30). We may assume that, afterthe second iteration, the soft estimates of transmitted symbolsare reliable, and hence, the interference cancellation is almostperfectly performed. In other words, we assume that si

k = sik,

and as a result, Λk,i = Λk,i. With this approximation, theexpression of wi

k in (30) simplifies to the following expression[17], [25]:

wik = h†

i

[hih

†i +

σ2z

σ2si

k

IMR

]−1

=1

h†ihi + σ2

z

σ2si

k

h†i (32)



where the filter coefficients no longer depend on k because σ2si

k

is constant for all k. Note that the last expression is nothingmore than a simple matched filter. In addition, note that, withthis approximation, we do not need to update the MMSE filterfor each sample time, because we effectively need to calculatewi

k once for a given realization of H, i.e., for the underlyingfading block. Furthermore, the matrix inversion in (30) isreplaced by a scalar inversion. Obviously, this simplificationcauses degradation to the receiver performance. However, thisperformance loss would be justified with regard to the complex-ity reduction, and hopefully, due to iterative detection, it wouldbe acceptable [26], [27].

Note that, unless otherwise mentioned, by turbo-PIC underperfect CSIR, we mean the exact formulation of the soft-PICdetector, i.e., (30).

C. Generalization to the Case of Linear Dispersion ST Codes

We have considered the simple spatial multiplexing (i.e.,V-BLAST) scheme at the transmitter. We show here that thedetectors’ formulations, which are presented for the case of spa-tial multiplexing, can also be applied to the more general caseof linear dispersion (LD) codes [28] with a slight modification.The reader is referred to [28] for more details on the LD codes’formulation.

Let s of dimension Q × 1 be the vector of data symbols priorto ST coding. By LD coding, these symbols are mapped intoa matrix X of dimension MT × Tu with Tu the number ofchannel uses. For an encoded matrix X , we receive the matrixY of dimension MR × Tu. To obtain a general formulation forthe receiver, we separate the real and the imaginary parts of theentries of s, Y , and X and stack them rowwise in vectors s, X,and Y of dimension 2Q × 1, 2MT Tu × 1, and 2MRTu × 1,respectively. Vectors s and Y are related through an equiv-alent channel matrix Heq of dimension (2MRTu × 2Q),which depends on H and the employed LD code (see [28]).We have

Y = Heq s + Z (33)

where Z is the vector of real AWGN of variance σ2z/2. We

see that, in the expressions of the detectors, we only have toconsider Heq, s, and Y, instead of H, s, and y, respectively.

V. IMPROVED DETECTION UNDER IMPERFECT CSIR

We propose here modifications to the MIMO detector tomitigate the impact of imperfect channel estimation on thereceiver performance. We first consider the MAP and, then, thesoft-PIC detector in the following sections. For soft-PIC, weconsider both cases of exact and simplified detectors.

A. Improved MAP Detection

We notice that the metric DML(sk,yk,H) involved in (23)requires the perfect channel matrix H, of which the receiver hassolely an estimate H. We propose to use the decoding metricDM(sk,yk, H) of (20) for the evaluation of the LLRs in (23).This step leads us to derive a new demapping rule adapted from

the imperfect CSIR. This way, we calculate L(dmk ) from (22),

where, for instance, the nominator is calculated as follows:

P(dm

k = 1|yk, H)

=∑

sk∈Sm1

exp{−DM(sk,yk, H)

}BMT∏n=1n �=m

P 1dec (dn

k )

=1

πeMR

∑sk∈Sm

1

1σ2

z + δσ2ε‖sk‖2

× exp

{‖yk − δHsk‖2

σ2z + δσ2

ε‖sk‖2

}BMT∏n=1n �=m

P 1dec (dn

k ) . (34)

B. Improved Soft-PIC Detection

As shown in (26) and (30), we need the channel H for bothinterference canceling and MMSE filtering. The receiver hasonly an imperfect channel estimate H; thus, the suboptimalmismatched solution consists of replacing Hi and hi in (26)and (30) by their estimates Hi and hi, respectively. As the firststep toward a realistic design, we make use of the availablechannel estimate H for interference cancellation. That is, (26)is rewritten as

yik

= yk − Hisik = his

ik + His

ik − His

ik + zk

for i = 1, . . . , MT (35)

where Hi is the (MR × (MT − 1)) matrix constructed fromH by discarding its ith column, i.e., hi. We now propose anovel improved PIC detector under imperfect CSIR. We notethat (35) naturally depends on the unknown channel matrix H,of which the receiver has only an imperfect estimate available.Instead of replacing the unknown channel by its estimate (i.e.,the mismatched approach), we use the posterior distribution(12) and make two modifications to the detector described inSection IV-B as follows.

The first modification that we propose concerns the designof the filter wi

k in (29). The cost function f(yk, sik,H) =

Esk,zk[|si

k − wikyi

k|2] is a function of the perfect channel H

(via yik); thus, we propose a modified filter wi

k, which waschosen to minimize the average of the mean-square error overall realizations of channel estimation errors. Using (12), wepropose the following filter design:

wik = arg min

wik∈CMR

EH,sk,zk


kyik

∣∣∣2∣∣∣∣ H]= arg min

w∈CMR

EH|H

[Esk,zk


kyik

∣∣∣2]] (36)

where, in the latter expression, we have assumed the indepen-dence between H, sk, and zk.

From (36) and after invoking the orthogonality principle[24], we obtain

wik=

(E

H|H

[Esk,zk

[si

kyik

†]])(E

H|H

[Esk,zk

[yi

kyi

k

†]])−1

.

(37)



After some algebraic manipulations in Appendix II-A, we getthe modified filter wi

k, directly as a function of Hi and hi, asfollows:

wik = Rsi

kyi

kR

−1yi

k

(38)

where

Rsikyi

k= δσ2

sikh†

i + (δ − 1)mk,iH†i (39)

with mk,i = siksi

k

†. δ is given by (13), and

Ryik

= δ2σ2si

khih

†i + δ2HiΛk,iH

†i + (δ2 − δ)himk,iH

†i

+ (δ2 − δ)Him†k,ih

†i + (1 − 2δ)HiΛk,iH

†i

+(σ2

z + (1 − δ)σ2si

k+ (1 − δ)tr(Λk,i)

)IMR

. (40)

To get more insight on the proposed detector, let us considerthe ideal case where perfect channel knowledge is available atthe receiver, i.e., H = H, and σ2

E = 0. We note that, in thiscase, δ = 1, and the posterior pdf (12) reduces to a Dirac deltafunction; consequently, the two filters wi

k and wik coincide.

Similarly, under near-perfect CSIR, which is obtained eitherwhen σ2

E → 0 or when NP → ∞, we have δ → 1, and thefilter wi

k gives a similar expression as wik in (30). However,

in the presence of estimation errors, the proposed improvedand mismatched detectors become different due to the inherentaveraging in (36), which provides a robust design that adaptsitself to the channel estimate available at the receiver.

Our second modification concerns the application of thederived filter wi

k to the received signal yik. By applying the

modified filter wik to yi

kin (35), we have

rik = wi

kyik

= wikhis

ik + wi

kHisik − wi

kHisik + wi

kzk.(41)

The latter equation is a function of the perfect channel of whichthe receiver has only an imperfect estimate. Again, insteadof replacing the perfect channel by its estimate (which is themismatched approach), we propose to average the filter outputrik by using the a posteriori distribution (12) as follows:

rik = E

H|H[rik

]= δwi

khi︸︷︷︸μk,i

sik + δwi

kHisik − wi

kHisik + wi

kzk︸︷︷︸ηk,i

(42)

where we have used Ehi|hi

[hi] = δhi, EHi|Hi

[Hi] = δHi, and

ηk,i is the interference-plus-noise ratio that affects the outputof the instantaneous MMSE filter ri

k. Based on (42), it is clearthat the output of the improved MMSE filter can be viewed asan equivalent AWGN channel with si

k at its input, i.e.,

rik = μk,is

ik + ηk,i. (43)

It is shown in [16] and [19] that, under perfect CSIR, a sim-ilar expression of ηk,i is well approximated by a zero-meanGaussian random variable with variance σ2

ηk,i. The parameters

μk,i and σ2ηk,i

are calculated at each time slot by using thesymbol statistics. The exact derivation of the variance σ2

ηk,iis

provided in Appendix II-B, for the more general case of theimproved detector.

Based on (42), we can calculate the LLRs on the correspond-ing bits of the detected symbols at the output of the MMSEfilter, which will be used by the SISO channel decoder. Let di,m

k

denote the mth (m = 1, . . . , B) bit that corresponds to sik. The

LLR on di,mk is given by

L(di,m

k

)= log

Pdem

(di,m

k = 1|rik, μk,i

)Pdem

(di,m

k = 0|rik, μk,i

)

= log

∑si

k∈Sm

1

exp{−|ri

k−μk,is

ik|2

σ2ηk,i

}B∏

n=1n �=m

P 1dec

(di,n

k

)∑

sik∈Sm

0

exp{−|ri

k−μk,isi

k|2

σ2ηk,i

}B∏

n=1n �=m

P 0dec

(di,n

k

) .

(44)

Note that¸ in contrast to the case of MAP detection where, in(23), Sm

1 and Sm0 are of size 2(MT B−1), here, the cardinality of

the sets Sm1 and Sm

0 is only equal to 2B−1.Finally, note that adopting the aforementioned improved

reception scheme does not considerably increase the complex-ity compared to the mismatched approach. More precisely,comparing the mismatched and the improved filter expressions[i.e., comparing (30) after replacing the perfect channel by itsestimate and (38)] shows that the improved filter requires onlya few number of extra matrix additions and multiplications,which does not have an important impact on the receivercomplexity; the required additional operations are one vectorby matrix multiplication and one vector addition for calcu-lating Rsi

kyi

k, as well as two matrix multiplications, two scalar

by matrix multiplications, and two matrix additions for calcu-lating Ryi

k.

1) Modification for the Simplified Soft-PIC Detector: Wehave presented a simplified formulation for the turbo-PIC de-tector; thus, in this section, we present the improved detectorformulation for the case of imperfect CSIR. For the first iter-ation, the formulation is like the case of exact soft-PIC andremains unchanged. For the next iterations, it can easily be veri-fied that replacing Λk,i by Λk,i in (38) does not, unfortunately,lead to a compact expression that involves a simple scalar in-version. However, with a slight modification in the interferencecancellation part, we can also derive a simple improved detectorsimilar to (32). This result is achieved by cancelling the residualinterference by using (26) instead of (35) as

yik

= hisik + His

ik − His

ik + zk. (45)

By following a similar procedure as in Section V-B, it can thenbe shown that the improved MMSE filter of (38) reduces to

wik = δh†

i

[δ2hih

†i + (1 − δ)

×(

1 +tr(Λk,i − Λk,i)

σ2si

k

+σ2

z

(1 − δ)σ2si

k

)

× IMR+

δ2Hi(Λk,i − Λk,i)H†i

σ2si

k

]−1

. (46)



Now, by setting Λk,i to be equal to Λk,i in (46), we obtainthe following simplified expression for the MMSE filter of theimproved detector:

wik = δh†

i

[δ2hih

†i + (1 − δ)IMR

+σ2

z

σ2si

k

IMR

]−1

=δ

δ2h†ihi + (1 − δ) + σ2

z

σ2si

k

h†i. (47)

The modification of the interference cancellation and thecalculation of the LLRs are the same, as provided by (42) and(44), respectively.

VI. NUMERICAL RESULTS

In this section, we provide some numerical results to showthe performance improvement by using the proposed modifieddetectors in the presence of channel-estimation errors. Theperformance is evaluated in terms of receiver bit error rates(BERs).

For channel coding in our BICM scheme, we consider therate 1/2 NRNSC code of constraint length K = 7, definedin octal form by (133, 171)8. Uncorrelated Rayleigh fadingchannel is considered, and each frame of symbols correspondsto Nc fading blocks. Channel coefficients are kept constantduring each fading block and are changed to new independentrealizations from one block to the next. For each fading block,we devote NP channel uses to the transmission of pilot se-quences. Throughout the simulations, each frame is composedof L = 128 channel uses for data symbols plus a total numberof NcNP channel uses for pilot transmission. Data symbolsbelong to QPSK or 16-QAM constellations with Gray labeling.We use mutually orthogonal QPSK pilot sequences for channelestimation, and the average pilot-symbol power is set to beequal to the average data-symbol power.

The interleaver is pseudorandom, which operates over theentire frame of size NI = LBMT bits (excluding pilots, ob-viously). Moreover, the number of receiver iterations is set to5. The signal-to-noise ratio (SNR) is considered in the form ofEb/N0 and includes the receiver antenna array gain MR. Weassume that the noise variance is known at the receiver.

A. Case of Turbo-MAP Detector

Let us first consider the turbo-MAP detector. Fig. 2 showsBER curves of the mismatched and improved receivers for thecase of QPSK modulation, MT = 2 and MR = 2, which wedenote by a (2 × 2) MIMO system. The number of channeluses for pilot transmission is NP ∈ {2, 4, 8}. As a reference,we have also presented the BER curve in the case of perfectCSIR. We notice that the required SNR to attain the BERof 10−5 with NP = 2 pilots is reduced by about 0.4 dB forthe improved detector compared with the mismatched detector.By increasing NP , the channel-estimation error becomes lessimportant, and the difference of the performances of the twodetectors decreases; the achieved gain in SNR at BER = 10−5

is about 0.2 and 0.1 dB for NP = 4 and NP = 8, respectively.

Fig. 2. BER performance of the improved and the mismatched turbo-MAPwith the 2 × 2 MIMO with the V-BLAST ST scheme, i.i.d. Rayleigh fadingwith Nc = 4, QPSK modulation, and training sequence length of NP ∈{2, 4, 8}.

Fig. 3. BER performance of the improved and the mismatched turbo-PIC withthe 2× 2) MIMO with the V-BLAST ST scheme, i.i.d. Rayleigh fading withNc = 4, QPSK modulation, and training sequence length of NP ∈ {2, 4, 8}.

In fact, the performance loss of the mismatched receiver withrespect to the improved receiver becomes insignificant forNP ≥ 8.

B. Case of the Turbo-PIC Detector

For the case of the turbo-PIC detector, Fig. 3 shows BERcurves of the mismatched and improved receivers under thesame conditions as in Fig. 2. We see that the gain in the SNR ofthe improved detector to attain the BER of 10−5 is now about1.4, 0.5, and 0.2 dB, respectively, for NP = 2, 4, and 8. In fact,the gain is more important than for turbo-MAP.

It is interesting to see the evolution of BER versus thenumber of iterations. We have shown in Fig. 4, for NP = 2,the corresponding BER curves for the cases of mismatched and



Fig. 4. BER performance of the improved and the mismatched turbo-PICthrough iterations. NP = 2. Other parameters are the same as in Fig. 3.Numbers on the curves indicate iteration numbers.

improved detectors for up to five iterations of the receiver. Wenotice that the two detectors have almost the same convergencetrend, and the major improvement is obtained after the seconditeration for both detectors. As a result, if, due to complexityreduction, we only process two receiver iterations, then we stillhave a considerable performance gain by using the improveddetector.

We have also considered the case of a 2×2 system with alarger sized signal constellation, i.e., the 16-QAM (results arenot shown due to space limits). The obtained gain by usingthe improved detector is less considerable compared with thecase of QPSK modulation. In fact, even with perfect CSIR, forlarger constellation sizes, turbo-PIC becomes more suboptimal,except in the presence of high (time or space) diversity. Conse-quently, the improved turbo-PIC does not offer a considerablegain compared to the mismatched detector.

C. Case of Turbo-PIC Detector With ST Coding

We have considered the simple spatial multiplexing at thetransmitter. It is interesting to study the performance of theimproved detector for more powerful ST codes. We considerhere as the ST scheme the optimized golden code (GLD),as presented in [30] for the case of two transmit antennasand MR ≥ MT , which offers full-rate and full-diversity withnonvanishing determinant. With this ST scheme, each vectorof four symbols s = [ s1 s2 s3 s4 ]T is mapped into a(2 × 2) matrix X as described as follows:

X =1√5

[α[s1 + θs2] α[s3 + θs4]γα[s3 + θs4] α[s1 + θs2]

](48)

where

θ =1 +

√5

2, α = 1 + j(1 − θ), θ = 1 − θ

α = 1 + j(1 − θ), γ = j, j =√−1.

Fig. 5. BER performance of the improved and the mismatched turbo-PIC withthe 2×2 MIMO with the GLD ST scheme, i.i.d. Rayleigh fading with Nc = 4,QPSK modulation, and training sequence length of NP ∈ {2, 4, 8}.

Fig. 6. BER performance of the exact and the simplified turbo-PIC imple-mentations with 2×2 MIMO, i.i.d. Rayleigh fading with Nc = 4, and per-fect CSIR.

Fig. 5 contrasts the BER curves of the receiver for the casesof improved and mismatched detectors. It is shown that theperformance gain by using the improved detector can be quiteconsiderable in this case. The SNR gains at a BER of 10−5 areof 2.3, 0.9, and 0.4 dB for NP = 2, 4, and 8, respectively.

D. Case of the Simplified Turbo-PIC Detector

Let us now consider the case of the simplified soft-PIC andstudy the performance gain by using the improved detector.Remember that, here, both the mismatched and the improvedsimplified detectors need to calculate the MMSE filter once perchannel realization and not for each channel use, as in the casefor the exact implementation of the soft-PIC.

First, in Fig. 6, we have compared the performances ofthe exact and the simplified turbo-PIC for the case of perfect



Fig. 7. BER performance of the improved and the mismatched simplifiedturbo-PIC with 2×2 MIMO with the V-BLAST ST scheme, i.i.d. Rayleighfading with Nc = 4, QPSK modulation, and training sequence length of NP ∈{2, 4, 8}.

CSIR, where a 2×2 system is considered with the simpleV-BLAST (QPSK modulation and 16-QAM) and with the GLDscheme (QPSK modulation). We notice that the performancedegradation by using the simplified soft-PIC detector at theBER of 10−5 is about 0.6 and 0.7 dB, respectively, for theV-BLAST and GLD schemes with a QPSK modulation andmore than 3 dB for the V-BLAST scheme with 16-QAM. In ef-fect, for large constellation sizes, the performance degradationby the simplification made in soft-PIC becomes considerable.

Now, consider Fig. 7, which compares the performances ofthe mismatched and the improved simplified detectors under thesame conditions as in Fig. 3, i.e., for a V-BLAST scheme withQPSK modulation. We notice that the gain in SNR by using theimproved detector is as important as in the case of the exactformulation of the soft-PIC and is about 1.4, 0.7, and 0.2 dB forNP = 2, 4, and 8, respectively.

VII. IMPACT ON THE CONVERGENCE OF TURBO-PIC

It is interesting to study the influence of the proposed modi-fication on the convergence of the turbo-PIC detector. For thispurpose, we consider the extrinsic-information transfer (EXIT)charts, which are simple efficient tools for the convergenceanalysis of iterative receivers [31], [32]. They are based on theflow of the extrinsic information exchanged between the SISOblocks in an iterative scheme and give insight into the conver-gence behavior of the receiver. In the following discussion, webriefly introduce this tool and refer to the provided referencesfor more details.

In the EXIT chart analysis, the LLRs input to a SISO blockare assumed to be uncorrelated and to follow a Gaussian dis-tribution, with its mean being related to its variance [31]. Notethat these assumptions are not perfectly satisfied in practice. Asa result, EXIT charts do not exactly predict the convergencebehavior of the receiver. For turbo-PIC, a posteriori (and notextrinsic) LLRs are fed from the decoder to the MIMO detector.

Fig. 8. EXIT charts analysis of the improved and the mismatched turbo-PICwith 2×2 MIMO with the GLD ST scheme, i.i.d. Rayleigh fading with Nc = 4,QPSK modulation, and training sequence length of NP = 2. Eb/N0 = 5 dB.“it” stands for iteration number.

However, it is quite logical to also consider the aforementionedassumptions for the a posteriori LLRs.

Let us use the subscripts .A (for a priori) and .E (forextrinsic) to denote the variables at the input and output of aSISO block, respectively. Note that, for the MIMO detector,the subscript .E corresponds to extrinsic LLRs, whereas for thedecoder, it corresponds to a posteriori LLRs. In addition, letus denote by IA and IE the mutual information at the inputand output of a SISO block, respectively. The behavior of theiterative detector is determined by associating IDET

E ⇒ IDECA

and, inversely, IDECE ⇒ IDET

A , where the superscripts .DET and.DEC refer to the MIMO detector and the channel decoder,respectively.

We have shown in Fig. 8 the EXIT charts of the soft-PICdetector for two cases of mismatched and improved receivers,as well as that of the Max-Log-MAP decoder.1 The exactformulation of the soft-PIC detector is considered. To better seethe difference between the two detectors, we have consideredthe GLD ST scheme and set NP to 2 and Eb/N0 to a relativelylow value, i.e., 5 dB. The iterative process is characterized by atrace between the EXIT curves of the MIMO detector and thedecoder. This is shown in the figure for the improved detector.Notice that, at the first iteration, where there is no a priori infor-mation on the transmitted symbols available, i.e., IDET

A = 0, weperform an MMSE detection, and the EXIT characteristic of thesoft-PIC is specified by a single point. For the next iterations,where we perform soft interference cancellation, the EXITcurve is obtained for 0 ≤ IDET

A ≤ 1. As a result, the EXIT tracestarts at the point that corresponds to the first iteration of thedetector and proceeds according to the curve corresponding tothe next iterations. Note that the obtained trajectory provides

1Note that it is not really correct to use the term “EXIT” chart for the SISOdecoder in turbo-PIC, because it delivers a posteriori LLRs at its output. Inaddition, note the particular form of the corresponding curve in Fig. 8, which isdue to the same reason.



an asymptotic (or an approximate) description of the receiverconvergence. In particular, for the assumption of uncorrelatedLLRs, this trajectory would correspond to the case of perfectinterleaving with infinite interleaver size.

We notice that the EXIT curve of the improved detector(for any iteration) lays above that of the mismatched detector.Interestingly, the distance between the curves of improved andmismatched detectors becomes much more considerable byincreasing the a priori information IDET

A . This result confirmsthe lower BER after convergence for the improved detector.Note that the final BER depends on the intersection point of theEXIT curves of the detector and the decoder. The closer thecorresponding IDEC

E is to one, the lower the BER becomes.The convergence rate is almost the same for the two detectors,because their EXIT curves have almost the same slope. Wenotice that both improved and mismatched detectors requireabout only two iterations to achieve the decoder output mutualinformation very close to one. In other words, by using theimproved soft-PIC detector, in practice, no improvement isachieved in terms of the receiver convergence rate. This resultis in accordance with the results in Fig. 4.

VIII. CONCLUSION AND DISCUSSIONS

By introducing a Bayesian approach that characterizes thechannel-estimation process, we have proposed a general de-tector design that takes into account the imperfect channelestimate. Using this design, we derived two improved iterativeMIMO receivers based on MAP and soft-PIC detectors thatmitigate the impact of channel uncertainty on the detectionperformance. The cases of the exact and the simplified im-plementations of the soft-PIC detector were both treated. Theformulation of the improved detector was provided for thesimple case of the V-BLAST scheme. We have also provided itsextension to the more general case of LD codes. We showed thatthe classically used mismatched detector becomes suboptimalonce it is compared with the proposed detectors, in particu-lar when only few pilots are devoted for channel estimation.The important point is that the performance improvement isobtained while imposing no considerable increase on the re-ceiver’s complexity. For ST schemes designed by imposingmore constraints on the coding rate and/or diversity, e.g.,the full-rate full-diversity GLD scheme, the receiver is moresensitive to the channel-estimation errors, and the improveddetector provides even more important gains. By analyzing theconvergence behavior of the iterative receiver, we showed thatthe improved soft-PIC detector does not bring a significantimprovement in the convergence rate compared with the mis-matched detector.

The amount of performance improvement provided by theimproved detector is a function of the channel-estimation errorvariance. In the presented results, we have noticed that a rela-tively small performance improvement is achieved, except fora very small number of pilot symbols. Thus, the proposed im-proved detector is particularly interesting under the conditionsof relatively fast fading. In such cases, it is important to transmitas few pilots as possible to avoid a significant loss in spectralefficiency. In other words, we can use few pilot symbols and

employ the improved detector, hence reducing the loss in datarate due to pilot insertion.

One should also note that, in the presented results, we haveset equal average power for pilot and data symbols. Obviously,if we allocate less power to pilots, the improved detector willprovide larger gains with respect to the mismatched detector.In other words, by employing the improved detector, we canreduce the pilot’s power and dedicate more power to datasymbols, hence improving the system performance in terms ofchannel capacity. Again, this result is of particular interest inrelatively fast-fading channels.

Finally, note that, although we have considered LS channelestimation in this work, the extension of the proposed detectorto the MMSE channel estimation is rather straightforward.

APPENDIX IA. Derivation of the a Posteriori Probability (10)

The following theorem is derived in [33].Theorem 1.1: Let x1 and x2 be circularly symmetric com-

plex Gaussian random vectors with zero means and full-rank covariance matrices Σij = E[xix

†j ]. Then, the conditional

random vector x1|x2 ∼ CN (μ,Σ) is circularly symmetriccomplex Gaussian with mean μ = Σ12Σ−1

22 x2 and covariancematrix Σ = Σ12Σ−1

22 Σ21.We denote by Hi and Hi the ith column of matrices H and

H, respectively, and set x1 = Hi and x2 = Hi. Based on (2)and (9), we have Σ11 = Σ12 = ΣH and Σ22 = ΣH + ΣE inTheorem 1.1. According to the theorem, the conditional pdfof Hi|Hi is a circularly symmetric complex Gaussian distribu-tion with

mean =ΣΔHi

whereΣΔ

Δ=ΣH(ΣH + ΣE)−1 (49)

covariance matrix =ΣH − ΣH(ΣH + ΣE)−1Σ†H

=ΣΔΣE . (50)

The equivalence in (50) is shown by left multiplying both sidesof (ΣE + ΣH) − Σ†

H = ΣE by ΣH(ΣE + ΣH)−1.Assuming the same covariance matrix for all columns of H

and H, we obtain the a posteriori pdf (10).

B. Evaluation of the Likelihood Function (17)

To evaluate the conditional expectation in (17), we use thefollowing theorem from [34].

Theorem 1.2: For a circularly symmetric complex randomvector u ∼ CN (m,Σ) with mean m = E[u], covariance ma-trix Σ = E[uu†] − mm†, and a Hermitian matrix A such thatI + ΣA > 0, we have

Eu

[exp{−u†Au}

]=

exp{−m†A(I + ΣA)−1m

}det{I + ΣA} . (51)

Let us define u = yk − Hsk. Using the a posteriori distribu-tion of (10) and after some algebra, we can derive the condi-tional pdf of u, given sk and H as u|(sk, H) ∼ CN (mu,Σu),



where mu = yk − ΣΔHsk, and Σu = ΣΔΣE‖sk‖2. We fur-ther define A = Σ−1

z . By applying Theorem 1.2, (17) is writtenas (52), shown at the bottom of page. Σz , ΣΔ and ΣE arediagonal matrices; thus, the latter equation is rewritten as

W (yk|H, sk)

=exp

{−(yk−δHsk)†

(Σz+δΣE‖sk‖2

)−1 (yk−δHsk)}

det {π (Σz+δΣE‖sk‖)}

= CN(δHsk,Σz+δΣcalE‖sk‖2

). (53)

APPENDIX IIDETAILS ON THE FORMULATION OF

THE IMPROVED SOFT-PIC

A. Derivation of the Improved MMSE Filter (38)

The inner expectations involved in (37) can easily be evalu-ated from (35) as

Rsikyi

k= Esk,zk

[si

kyik

†] = σ2si

kh†

i + mk,i(Hi − Hi)† (54)

Ryik

= Esk,zk

[yi

kyi

k

†]=σ2

sikhih

†i + HiΛk,iH

†i + himk,iH

†i + Him

†k,ih

†i

− himk,iH†i − Him

†k,ih

†i−HiΛk,iH

†i−HiΛk,iH

†i

+ HiΛk,iH†i + σ2

zIMR. (55)

Thus, we have to evaluate the outer expectations as follows:

Rsikyi

k= E

H|H[Rsikyi

k]

Ryik

= EH|H[Ryi

k].

To compute for the aforementioned expectations, we use thefollowing lemma [35].

Lemma 1: For a circularly symmetric complex Gaussianrandom rowwise vector x ∼ CN (μ,Σ) and a Hermitianmatrix A, we have

Ex[xAx†] = tr(AΣ) + μAμ†. (56)

By applying Lemma 1 and using the a posteriori channel pdf(12), it is straightforward to find Rsi

kyi

kin (39) as

Rsikyi

k= E

H|H

[Rsi

kyi

k

]= δσ2

sikh†

i + (δ − 1)mk,iH†i. (57)

The evaluation of Ryik

in (40) involves the following equalities:

Ehi|hi

[hih

†i

]= δ2hih

†i + (1 − δ)IMR

Ehi|hi

[himk,iH

†i

]= δ2himk,iH

†i

EH

i|H

i

[HiΛk,iH

†i

]= δHiΛk,iH

†i

EH

i|H

i

[HiΛk,iH

†i

]= δ2HiΛk,iH

†i + (1 − δ)tr(Λk,i)IMR

.

Now, by using the aforementioned equalities, we obtain

Ryik= δ2σ2

sikhih

†i + (1− δ)σ2

sikIMR

+ σ2zIMR

+δ2HiΛk,iH†i

+ (1 − δ)tr(Λk,i)IMR+ δ2himk,iH

†i + δ2Him

†k,ih

†i

− δhimk,iH†i − δHim

†k,ih

†i − δHiΛk,iH

†i

− δHiΛk,iH†i + HiΛk,iH

†i. (58)

Equation (40) directly follows after rearranging the terms in(58). As a result, based on (57) and (58), we obtain the im-proved MMSE filter (38).

B. Derivation of the Variance σ2ηk,i

in (44)

Based on (27), we can evaluate the mean-square error (MSE)at the output of the turbo-PIC detector as

σ2MSE = E


kyik

∣∣∣2]=σ2

sik− Rsi

kyi

kwi

k† − wi

kRyiksi

k+ wi

kRyikwi

k†. (59)

wik = Rsi

kyi

kR−1

yik

; thus, we have Rsikyi

kwi†

k = wikRyi

kwi†

k .

Consequently, (59) reduces to

σ2MSE = σ2

sikwi

kRyiksi

k. (60)

For the case of the improved detector, after using wik and Rsi

kyi

k

from (38) and (39) instead of wik and Rsi

kyi

kin (60), we obtain

σ2MSE-IM = σ2

sik(1 − μk,i) − (δ − 1)wi

kHim†k,i (61)

where μk,i = δwikhi.Alternatively, based on (43), we have

σ2MSE-IM = E

[∣∣sik −

(μk,is

ik + ηk,i

)∣∣2]= (1 − μk,i)2σ2

sik

+ σ2ηi

k− 2Re

×((δ − 1)

(1 − μ∗

k,i

)wi

kHim†k,i

). (62)

W (yk|H, sk) = EH|H

[exp

{−(yk − Hsk)†Σ−1

z (yk − Hsk)}

det{πΣz}

]

=exp

{−(yk − ΣΔHs)†Σ−1

z

(I + ΣΔΣE‖sk‖2Σ−1

z

)−1 (yk − ΣΔHsk)}

det(πΣz

(I + ΣΔΣE‖sk‖2Σ−1

z

)) (52)



Comparing (61) and (62) leads to

σ2ηi

k=

(μk,i − μ2

k,i

)σ2

sik

+ (δ − 1)

×[2Re

((1 − μ∗

k,i

)wi

kHim†k,i

)− wi

kHim†k,i

]. (63)

Notice that, by setting δ to be equal to one (which correspondsto perfect CSIR) in (63), we retrieve the classical expressionderived in the literature, i.e., σ2

ηik

= (μk,i − μ2k,i)σ

2si

k

(for ex-

ample, see [16] and [29]).

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Seyed Mohammad-Sajad Sadough (S’04–M’08)was born in Paris, France, in 1979. He received theB.Sc. degree in electrical engineering (electronics)from the Shahid Beheshti University, Tehran, Iran,in 2002 and the M.Sc. and Ph.D. degrees in elec-trical engineering (telecommunications) from ParisSud 11 University, Orsay, France, in 2004 and 2008,respectively.

From 2004 to 2007, he was with the National En-gineering School in Advanced Techniques (ENSTA),Paris, and the Laboratory of Signals and Systems,

Supélec—National Scientific Research Center (CNRS), Gif sur Yvette, France.He was a Lecturer with the Department of Electronics and Computer En-gineering, ENSTA, where his research activities were focused on improvedreception schemes for ultrawideband communication systems. From December2007 to September 2008, he was a Postdoctoral Researcher with the labora-toire des Signaux et Systèmes (LSS), Supélec-CNRS, where he was involvedin research projects with Alcatel-Lucent on satellite mobile communicationsystems. Since October 2008, he has been with the Faculty of Electrical andComputer Engineering, Shahid Beheshti University, where he is currently anAssistant Professor with the Department of Telecommunications. His researchinterests include signal processing for wireless communications, with particularemphasis on multicarrier and multiple-input–multiple-output systems, jointchannel estimation and decoding, iterative reception schemes, and interferencecancellation under partial channel-state information.



Mohammad-Ali Khalighi (SM’07) received thePh.D. degree in electrical engineering from the Insti-tut National Polytechnique de Grenoble, Grenoble,France, in 2002.

From 2002 to 2005, he was with Grenoble ImagesParole Signal Automatique (GIPSA)-Lab, the EcoleNationale Superieure des Télécommunications—Telecom ParisTech, Paris, France, and the Institutd’Électronique et de Télécommunications de RennesLaboratory, Rennes, France, as a Postdoctoral Re-search Fellow. Since 2005, he has been with the

École Centrale Marseille and the Institut Fresnel, Marseille, France, where heis currently an Assistant Professor. His research interests include coding, signaldetection, and channel estimation for high-data-rate communication systems.

Pierre Duhamel (M’87–SM’87–F’98) was born inFrance in 1953. He received the B.Eng. degree inelectrical engineering from the National Institutefor Applied Sciences, Rennes, France, in 1975 andthe Dr.Eng. and Doctorat ès Sciences degrees fromOrsay University, Orsay, France, in 1978 and 1986,respectively.

From 1975 to 1980, he was with Thomson-CSF,Paris, France, working on circuit theory and sig-nal processing, including digital filtering and analogfault diagnosis. In 1980, he joined the National Re-

search Center in Telecommunications, Issy les Moulineaux, France, where hisresearch activities were first concerned with the design of recursive charged-coupled device filters. Later, he worked on fast algorithms for computingFourier transforms and convolutions and applied similar techniques to adaptivefiltering, spectral analysis, and wavelet transforms. From 1993 to September2000, he was a Professor with the National School of Engineering in Telecom-munications, Paris, where his research activities focused on signal processingfor communications, and where he was the Head of the Signal and ImageProcessing Department from 1997 to 2000. He is currently with the NationalScientific Research Centre/Laboratory of Signals and Systems—Supélec, Gifsur Yvette, France, where he is developing studies in signal processing for com-munications (including equalization, iterative decoding, multicarrier systems,and cooperation) and signal/image processing for multimedia applications,including source coding, joint source/channel coding, watermarking, and audio-processing. He is currently investigating the application of recent informationtheory results to communication theory.

Dr. Duhamel was the Chair of the Digital Signal Processing Committee from1996 to 1998 and a member of the signal processing for Com Committee until2001. He was an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL

PROCESSING from 1989 to 1991, an Associate Editor for the IEEE SIGNAL

PROCESSING LETTERS, and a Guest Editor of the IEEE TRANSACTIONS ON

SIGNAL PROCESSING Special Issue on Wavelets. He was a Distinguished Lec-turer of the IEEE in 1999 and was a General Cochair of the 2001 InternationalWorkshop on Multimedia Signal Processing, Cannes, France. He was also aTechnical Cochair of the 2006 IEEE International Conference on Acoustics,Speech, and Signal Processing, Toulouse, France. He is a coauthor of the paperon subspace-based methods for blind equalization, for which he received theBest Paper Award from the IEEE TRANSACTIONS ON SIGNAL PROCESSING

in 1998. He received the Grand Prix France Telecom from the French ScienceAcademy in 2000.


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